Texture Paint won't paint on some areas
The parts that can't be painted on have inverted normals, so the normals are facing instead of out. Add to that the "Occlude" option in project paint is enabled. To fix this, go into Object mode and select all the faces and press Ctrl+N to correct the normals, or if they are supposed to be that way, look under the Options tab of the Tool shelf in the Project Paint panel and disable the "Occlude" option.